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It is widely known that the twisted bilayer graphene (TBG) shows flat bandsat magic angles, which can be well described by the effective continuum modelderived by Bistritzer and MacDonald (BM). We propose in this paper a similartwisted bilayer system but defined on the square lattice with $\pi$-flux perplaquette, and study its spectrum using the BM Hamiltonian with a mass termwhich is originated from the staggered potential. The basic difference betweenthe TBG and the present model is simply rotational symmetry, $C_3$ vs $C_4$, aswell as a mass term. Nevertheless, the feature of the flat bands is quitedifferent: Those of the TBG appear at magic angles only, while the presentmodel shows many flat bands, which are reminiscent of Landau levels, quitestably at any angles even in the absence of a magnetic field other than$\pi$-flux which keeps time reversal (TR) symmetry. Moreover, flat bands emergein the mass gap of the Dirac spectrum, and each state composing these flatbands is well-localized at the position forming the moir\'e lattice. It turnsout that the moir\'e potential serves as a periodic magnetic field, which cangive energies smaller that the gap around moir\'e lattice positions. We derivea local Hamiltonian valid around the moir\'e lattice sites and show that itindeed reproduces the energies of the flat bands within the mass gap. Sincethese mid-gap states are localized at the moir\'e lattice, they form degeneratelevels, which may be referred to as moir\'e Landau levels, although themechanism of degeneracies are different from the conventional Landau levels.Interestingly, doubled fermions of the BH Hamiltonian associated with twolayers have opposite charges when they couple with the effective moir\'emagnetic filed, which concern TR symmetry.

Moir\'e Landau levels of a $C_4$-symmetric twisted bilayer system in the absence of a magnetic field